The following is an outline of this article. In Section REF , we summary some basic results related to the rigid partition in [1]}.
In Section REF , we introduce the definition and the construction of symbol in [2]}. Then we present the construction of the representative element \((\lambda ^{^{\prime }}, \lambda ^{^{\prime \prime }})_R\)  of rigid surface operators with the given symbol.
In Section REF , we give the constructions of the fingerprint invariant in [3]}. We calculate the fingerprint of the representative element \((\lambda ^{^{\prime }}, \lambda ^{^{\prime \prime }})_R\) . We also discuss the construction of the representative element \(\mu _r\)  of \(\lambda ^{^{\prime }}+ \lambda ^{^{\prime \prime }}\)  with the fingerprint given. our construction can be regarded as the maps between the fingerprint invariant and symbol invariant.
In Section REF , we discuss the realization of the above results for the \(C_n\)  and \(D_n\)  theories.
